## Abstract

We prove a generalization to the totally real field case of the Waldspurger's formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger's formula as a combination of two ingredients - an equality between global distributions, and a dichotomy result for theta correspondence. As applications we generalize the Kohnen-Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindelöf hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting.

Original language | English (US) |
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Pages (from-to) | 333-384 |

Number of pages | 52 |

Journal | Geometric and Functional Analysis |

Volume | 17 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2007 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology

## Keywords

- Half integral weight forms
- Special values of L-functions
- Waldspurger correspondence